Wave-Function Expansion in the Brillouin Zone: Silicon

A. James Hughes

doi:10.1103/PhysRev.166.776

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Wave-Function Expansion in the Brillouin Zone: Silicon

A. James Hughes^*

Applied Research Laboratory, Aeronutronic Division of Philco-Ford Corporation, Newport Beach, California

^*Present address: Theoretical Physics Department, Research and Engineering Division, Autonetics, Anaheim, Calif. 92803.

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Phys. Rev. 166, 776 – Published 15 February, 1968

DOI: https://doi.org/10.1103/PhysRev.166.776

Abstract

Wave functions are required throughout the Brillouin zone for the treatment of many problems in solid-state physics. It therefore is of interest and considerable importance to develop analytical representations of the k dependence of wave functions throughout the Brillouin zone. In this paper the momentum wave functions $b_{n} (k, K_{s})$ in the plane-wave expansion of the Bloch function $ψ_{n} (k, r)$ are considered. An analytical representation for the $b_{n} (k, K_{s})$ based on a set of symmetrized polynomials is proposed. The specific polynomials, although they pertain to general points in the Brillouin zone, are somewhat analogous to the Kubic harmonics employed in the cellular method. A primary distinction, however, is that while formerly one was concerned with space-group operations in the group of the k vector, here we are concerned with operations in the group of the $K_{s}$ vector for which $K_{s}$ in $b_{n} (k, K_{s})$ is left invariant.

References (12)

D. Brust, Phys. Rev. 134, A1337 (1964)
N. Bloembergen, R. K. Chang, and J. Ducuing, in Physics of Quantum Electronics, edited by P. L. Kelly, B. Lax, and P. E. Tannewald (McGraw-Hill Book Publishing Co., Inc., New York, 1966), p. 67
J. Callaway and A. J. Hughes, Phys. Rev. 156, 860 (1967)
F. C. von der Lage and H. A. Bethe, Phys. Rev. 71, 612 (1947)
D. G. Bell, Rev. Mod. Phys. 26, 311 (1954)
R. S. Knox and A. Gold, Symmetry in the Solid State (W. A. Benjamin, Inc., New York, 1964), p. 41
J. Callaway, Energy Band Theory (Academic Press Inc., New York, 1964), p. 23
J. Callaway, in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic Press Inc., New York, 1958), Vol. 7, p. 99
E. O. Kane, in Semiconductors and Semimetals, edited by R. K. Willardson and A. C. Beer (Academic Press Inc., New York, 1966), Vol. 1, p. 75
L. I. Schiff, Quantum Mechanics (McGraw-Hill Book Publishing Co., Inc., New York, 1955), p. 154
L. Mariot, Group Theory and Solid State Physics (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962), p. 86
J. Callaway and A. James Hughes (unpublished)

[c1] D. Brust, Phys. Rev. 134, A1337 (1964)

[c2] N. Bloembergen, R. K. Chang, and J. Ducuing, in Physics of Quantum Electronics, edited by P. L. Kelly, B. Lax, and P. E. Tannewald (McGraw-Hill Book Publishing Co., Inc., New York, 1966), p. 67

[c3] J. Callaway and A. J. Hughes, Phys. Rev. 156, 860 (1967)

[c4] F. C. von der Lage and H. A. Bethe, Phys. Rev. 71, 612 (1947)

[c5] D. G. Bell, Rev. Mod. Phys. 26, 311 (1954)

[c6] R. S. Knox and A. Gold, Symmetry in the Solid State (W. A. Benjamin, Inc., New York, 1964), p. 41

[c7] J. Callaway, Energy Band Theory (Academic Press Inc., New York, 1964), p. 23

[c8] J. Callaway, in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic Press Inc., New York, 1958), Vol. 7, p. 99

[c9] E. O. Kane, in Semiconductors and Semimetals, edited by R. K. Willardson and A. C. Beer (Academic Press Inc., New York, 1966), Vol. 1, p. 75

[c10] L. I. Schiff, Quantum Mechanics (McGraw-Hill Book Publishing Co., Inc., New York, 1955), p. 154

[c11] L. Mariot, Group Theory and Solid State Physics (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962), p. 86

[c12] J. Callaway and A. James Hughes (unpublished)

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